Find the principal and general solutions of the following equations:

Q (4)  \small cosec x = -2

Answers (1)
S seema garhwal

We know that
                    cosec \frac{\pi}{6} = 2
                  
                   cosec (\pi + \frac{\pi}{6}) = -cosec\frac{\pi}{6} = -2               and also                cosec (2\pi - \frac{\pi}{6}) = cosec\frac{11\pi}{6} = -2
So,
                   cosec x= cosec\frac{7\pi}{6}                                  and                                   cosec x= cosec\frac{11\pi}{6}

So, the principal solutions are x = \frac{7\pi}{6} \ and \ \frac{11\pi}{6} 
 

 

Now,
             cosec x= cosec\frac{7\pi}{6}
             
              \sin x = \sin\frac{7\pi}{6}                                                                                     \left ( \because \sin x = \frac{1}{cosec x} \right )

          x = n\pi + (-1)^{n}\frac{7\pi}{6}
Therefore, the general solution is

  x = n\pi + (-1)^{n}\frac{7\pi}{6}     

where n \ \epsilon \ Z

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